Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weisfeiler and Lehman Go Categorical

Seongjin Choi, Gahee Kim, Se-Young Yun

TL;DR

This work introduces the CatWL framework to unify higher-order lifting across data types by lifting objects into graded posets via functors, thereby enabling isomorphism-aware learning beyond standard WL. It formalizes a graded-WL refinement and shows how to instantiate neural architectures, termed F-CatMPN, through two principal functors for hypergraphs: the Incidence Poset Functor $I$ and the Symmetric Simplicial Complex Functor $S$. The authors prove that both $I$-CatWL and $S$-CatWL are not less powerful than HWL, and they validate these claims with extensive experiments on six real-world hypergraph benchmarks, finding that $I$-HIN excels on dense topologies while $S$-HIN shines on structure-rich, sparse hypergraphs. The results demonstrate the practical impact of category-theoretic lifting as a design principle for expressive, geometry-aware hypergraph learning, providing a scalable pathway to unify higher-order graph representations across modalities.

Abstract

While lifting map has significantly enhanced the expressivity of graph neural networks, extending this paradigm to hypergraphs remains fragmented. To address this, we introduce the categorical Weisfeiler-Lehman framework, which formalizes lifting as a functorial mapping from an arbitrary data category to the unifying category of graded posets. When applied to hypergraphs, this perspective allows us to systematically derive Hypergraph Isomorphism Networks, a family of neural architectures where the message passing topology is strictly determined by the choice of functor. We introduce two distinct functors from the category of hypergraphs: an incidence functor and a symmetric simplicial complex functor. While the incidence architecture structurally mirrors standard bipartite schemes, our functorial derivation enforces a richer information flow over the resulting poset, capturing complex intersection geometries often missed by existing methods. We theoretically characterize the expressivity of these models, proving that both the incidence-based and symmetric simplicial approaches subsume the expressive power of the standard Hypergraph Weisfeiler-Lehman test. Extensive experiments on real-world benchmarks validate these theoretical findings.

Weisfeiler and Lehman Go Categorical

TL;DR

This work introduces the CatWL framework to unify higher-order lifting across data types by lifting objects into graded posets via functors, thereby enabling isomorphism-aware learning beyond standard WL. It formalizes a graded-WL refinement and shows how to instantiate neural architectures, termed F-CatMPN, through two principal functors for hypergraphs: the Incidence Poset Functor and the Symmetric Simplicial Complex Functor . The authors prove that both -CatWL and -CatWL are not less powerful than HWL, and they validate these claims with extensive experiments on six real-world hypergraph benchmarks, finding that -HIN excels on dense topologies while -HIN shines on structure-rich, sparse hypergraphs. The results demonstrate the practical impact of category-theoretic lifting as a design principle for expressive, geometry-aware hypergraph learning, providing a scalable pathway to unify higher-order graph representations across modalities.

Abstract

While lifting map has significantly enhanced the expressivity of graph neural networks, extending this paradigm to hypergraphs remains fragmented. To address this, we introduce the categorical Weisfeiler-Lehman framework, which formalizes lifting as a functorial mapping from an arbitrary data category to the unifying category of graded posets. When applied to hypergraphs, this perspective allows us to systematically derive Hypergraph Isomorphism Networks, a family of neural architectures where the message passing topology is strictly determined by the choice of functor. We introduce two distinct functors from the category of hypergraphs: an incidence functor and a symmetric simplicial complex functor. While the incidence architecture structurally mirrors standard bipartite schemes, our functorial derivation enforces a richer information flow over the resulting poset, capturing complex intersection geometries often missed by existing methods. We theoretically characterize the expressivity of these models, proving that both the incidence-based and symmetric simplicial approaches subsume the expressive power of the standard Hypergraph Weisfeiler-Lehman test. Extensive experiments on real-world benchmarks validate these theoretical findings.
Paper Structure (47 sections, 15 theorems, 12 equations, 4 figures, 4 tables)

This paper contains 47 sections, 15 theorems, 12 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Why Category Theory
  3. The Lifting Mismatch
  4. A Universal domain
  5. Functorial Lifting
  6. Categorical WL test
  7. Graded WL Test
  8. Categories and Functors
  9. Categorical WL Test
  10. Hypergraph Isomorphism Networks
  11. Categorical Message Passing Networks
  12. Two Architectures from Two Functors
  13. The Incidence Poset Functor ($I$)
  14. The Symmetric Simplicial Complex Functor ($S$)
  15. Related work
  16. ...and 32 more sections

Key Result

theorem 1

Let $F: \mathcal{C} \to \mathbf{Poset}$ be a functor. The $F$-CatWL is at least as powerful as the $F$-CatMPN. If the update and aggregation functions are injective and the network has sufficient depths and widths, the $F$-CatMPN has the same expressive power as $F$-CatWL.

Figures (4)

  • Figure 1: $I$ is a functor from the category of hypergraphs to the category of graded posets by sending a hypergraph to its incidence poset. $I$ send a morphism of hypergraphs $f$ to a morphism of graded posets $I(f)$ and $I$ preserves the identity morphism. This implies $I$ is a lifting map.
  • Figure 2: Visualization of a single GWL refinement step on a sample graded poset $P$. The Hasse diagram depicts the covering relations $\prec$, where arrows point from covered elements to covering elements (e.g., $\sigma_4 \prec \sigma$). Nodes are colored according to the coloring $c$, and the 4-adjacency multisets for the element $\sigma$ are explicitly listed.
  • Figure 3: Comparison of lifting functors on a sample hypergraph $H$ (left). The incidence poset $I(H)$ (center) treats hyperedges as dimension 1 objects, while the symmetric simplicial complex $S(H)$ (right) explicitly represents all sub-relations.
  • Figure 4: Two non-isomorphic hypergraphs $H$ and $H'$ that are indistinguishable by HWL test but distinguished by $I$-CatWL test and $S$-CatWL test.

Theorems & Definitions (38)

  • definition 1: Graded Poset
  • definition 2: Graded Weisfeiler-Lehman (GWL)
  • definition 3: Category
  • definition 4: Functor
  • definition 5: CatWL
  • definition 6: $F$-CatMPN
  • theorem 1: Expressivity of $F$-CatMPN
  • definition 7: Incidence Poset Functor
  • theorem 2
  • definition 8: Symmetric Simplicial Complex Functor $S$
  • ...and 28 more